PHYSICS OF FLUIDS
VOLUME 11, NUMBER 2
FEBRUARY 1999
Streamline topologies near simple degenerate critical points
in two-dimensional flow away from boundaries
Morten Bro” ns and Johan Nicolai Hartnack
Department of Mathematics, Technical University of Denmark, Building 303, DK-2800 Lyngby, Denmark
~Received 16 April 1998; accepted 23 October 1998!
Streamline patterns and their bifurcations in two-dimensional incompressible flow are investigated
from a topological point of view. The velocity field is expanded at a point in the fluid, and the
expansion coefficients are considered as bifurcation parameters. A series of nonlinear coordinate
changes results in a much simplified system of differential equations for the streamlines ~a normal
form! encapsulating all the features of the original system. From this, we obtain a complete
description of bifurcations up to codimension three close to a simple linear degeneracy. As a special
case we develop the theory for flows with reflectional symmetry. We show that all the patterns
obtained can be realized in steady Navier–Stokes or Stokes flow, but an unresolved difficulty arises
in the symmetric case for Navier–Stokes flow. The theory is applied to the patterns and bifurcations
found numerically in two recent studies of Stokes flow in confined domains. © 1999 American
Institute of Physics. @S1070-6631~99!02102-9#
ric flow,18,19 and the streamline topology of point vortex
flow.20 In visualization experiments a topological approach
has been used for pattern recognition21 and for constructing a
wavelet basis for structure identification.22
When the flow field changes due to unsteadiness of the
flow or an external change of parameters or boundary conditions, the streamline topology may also change. When the
flow field is varied through a degenerate configuration that is
structurally unstable, the streamline pattern can bifurcate and
new structures arise. Such bifurcations for 2-D flow close to
a planar wall and in conical flow are studied by Bakker.23
Our aim in the present study is the same, but for 2-D flow
away from boundaries.
To obtain local information about a velocity field close
to a critical point, we use a Taylor expansion of the streamfunction. The coefficients in the Taylor series are then considered as bifurcation parameters. Even for low-order expansions, the number of coefficients is large, and the bifurcation
analysis is complicated. However, by a systematic use of
normal form transformations,24 the number of free parameters is reduced significantly. This technique was used by
Bro” ns18 for axisymmetric flow and by Hartnack25 to describe
flow close to fixed, possibly curved, walls.
We will classify the possible degenerate patterns that
occur with a simple degeneracy in the linear part and then
obtain their unfoldings, that is, all possible patterns that can
occur when such a configuration is perturbed slightly. A degeneracy is characterized by its codimension, which is the
minimal number of parameters needed to describe the unfolding. From the normal form analysis the codimension is
readily determined, and we show the associated bifurcation
diagrams of codimension up to three. As a special case we
develop the theory for flow with reflectional symmetry.
We also show that the constraints imposed by the vorticity transport equation for steady Navier–Stokes or Stokes
flow has no consequence for the streamline topology in the
I. INTRODUCTION
A number of fundamental concepts used to describe fluid
flows involve local properties of the streamlines. Think, for
example, of vortices and eddies, stagnation points, dividing
streamlines, recirculating zones, and the process of separation. Even if some of these phenomena do not have an exact
definition in terms of streamlines ~and even no formal definition at all1!, streamline patterns always play a role in identifying important properties of a flow. In the present paper
we examine streamline patterns and their bifurcations from a
topological point of view, using methods from nonlinear dynamics.
Given a velocity field v~x! at some time instant t 0 , the
streamlines are found as trajectories of the ordinary differential equations ẋ5v(x), a nonlinear dynamical system in dimension two or three. Low-dimensional nonlinear dynamics
is concerned exactly with the topology of the trajectories of
such equations and hence provides a coherent theoretical
framework as well as a terminology for fluid structures. For
example, a stagnation point ~where v50! is known in dynamics as a critical point, and for two-dimensional ~2-D!
incompressible flow there are two nondegenerate possibilities: If the point is a center, the fluid mechanics interpretation
is a vortex or eddying motion, while a saddle represents a
point of stagnation where the separatrices are the dividing
streamlines.
The qualitative study of streamlines using tools and
ideas from nonlinear dynamics has a long and scattered history in fluid mechanics; see, e.g., the review papers.2,3 Early
work on separation was performed by Legendre,4
Oswatitsch,5 Davey,6 and Lighthill.7 Hunt et al.8 studied
flow around obstacles and Dallmann9–11 have considered
both separation and vortex structures. Other applications are
wake flow behind bluff bodies,12–14 flow close to free and
viscous surfaces,15,16 channel flow over a step,17 axisymmet1070-6631/99/11(2)/314/11/$15.00
314
© 1999 American Institute of Physics
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M. Bro” ns and J. N. Hartnack
Phys. Fluids, Vol. 11, No. 2, February 1999
general case. In the symmetrical case the same holds for
Stokes flow, but a structural difference leaves the problem
open for Navier–Stokes flow.
Our main purpose in this paper is to establish a theoretical framework for analyzing flow topologies and their metamorphoses. To illustrate its scope we conclude by indicating
how the theory can be applied to understand the streamline
patterns and bifurcations recently observed numerically in
two specific Stokes flows: the flow in a rectangular cavity26
and the flow between rotating coaxial cylinders.27
315
FIG. 1. Nondegenerate critical points. ~a! Center; ~b! saddle.
showing that the type of the critical point depends on the
angle between the gradients of the velocity components at
the point.
II. GENERAL FLOW PATTERNS
A. Taylor expansions and regular critical points
B. Degenerate critical points and normal forms
We consider an incompressible two-dimensional flow far
from any boundaries. A streamfunction c exists such that the
streamlines are found from
If u J u 50, the critical point is degenerate, and higherorder terms become decisive for the streamline pattern.
There are two distinct subcases. Either zero is an eigenvalue
with geometric multiplicity one, or J is the zero matrix. In
the present paper we consider the first case only, which we
will refer to as a simple linear degeneracy. The latter case is
considerably more difficult and we briefly touch upon it in
Sec. IV and Sec. VI. In terms of physical quantities we have
ẋ5u5
]c
,
]y
ẏ5 v 52
]c
.
]x
~1!
To obtain local information of the flow close to a given point
that we take to be the origin, c is expanded in a power series,
`
c5
(
i1 j51
~2!
a i, j x i y j .
For the arguments in this section, the analyticity of c is not
important. We will impose certain nondegeneracy conditions
on the coefficients at some finite order, and the results we
obtain only depend on c being continuously differentiable to
that order.
The expansion coefficients in Eq. ~2! have a direct physical interpretation. The first-order coefficients are related to
the components of the velocity vector at the origin,
a 0,15u ~ 0,0! ,
a 1,052 v~ 0,0! ,
~3!
and the higher-order coefficients are related to the viscous
stress tensor and its derivatives evaluated at the origin. For
example,
t xx 52 t y y 52 m
t xy 5 t yx 5 m
S
]u
52 m a 1,1 ,
]x
~4a!
D
]u ]v
1
52 m ~ a 0,22a 2,0! .
]y ]x
~4b!
The linear approximation of the equations for the
streamlines is
SD S D S
a 1,1
2a 0,2
ẋ
a 0,1
5
1
2a 1,0
ẏ
22a 2,0 2a 1,1
DS D
x
.
y
~5!
If a 1,05a 0,150 the origin is a critical point. The streamline
pattern close to the origin is determined using standard
theory for Hamiltonian systems. If the determinant of the
Jacobian matrix u J u 54a 2,0a 0,22a 21,1 is positive, the critical
point is a center, if uJu is negative it is a saddle. See Fig. 1. In
terms of physical quantities, we have
]u ]v ]u ]v
d –“ v , ~6!
1
5“u
u J u 54a 2,0a 0,22a 21,152
]y ]x ]x ]y
d –“ v 50,
“u
with u “u u 1 u “ v u Þ0.
~7!
Hence, in a simple linear degenerate critical point the gradients of u and v are parallel or at most one of them is the zero
vector.
Without loss of generality, we can assume that the coordinate system is chosen such that
a 1,15a 2,050,
but a 0,2Þ0.
~8!
Before analyzing the system ~1! we will make a series of
normal form transformations24 to successively simplify quadratic, cubic, and higher-order terms. The system is Hamiltonian and we wish to preserve this property under the transformation. Among all transformations preserving the
Hamiltonian structure canonical transformations have been
widely used in analytical mechanics. For a one-degree-offreedom system as considered here, a canonical transformation is simply an area-preserving map of the phase space ~the
physical flow plane!.
Canonical transformations are easily found via generating functions. Denote the new coordinates ~j,h!. For a
smooth function S(y, j ) consider the equations
x5
]S
,
]y
h5
]S
.
]j
~9!
If these can be solved for ~x,y!, the solution defines a canonical transformation from new to old coordinates, x5x( j , h ),
y5y( j , h ) ~Ref. 28, Chap. 8!.
To simplify the quadratic terms of Eqs. ~1!—cubic terms
of c—the normal form procedure uses a transformation that
is the identity plus quadratic terms, and, possibly, higherorder terms as well. We choose a generating function,
S5y j 1
(
i1 j53
s i, j y i j j .
~10!
With this, Eqs. ~9! become
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316
M. Bro” ns and J. N. Hartnack
Phys. Fluids, Vol. 11, No. 2, February 1999
x5 j 13s 3,0y 2 12s 2,1y j 1s 1,2j 2 ,
~11a!
h 5y1s 2,1y 2 12s 1,2y j 13s 0,3j 2 .
~11b!
We look for solutions to Eqs. ~11! in the form of Taylor
series,
`
x5 j 1
`
(
a i, j j i h j ,
i1 j52
y5 h 1
(
i1 j52
b i, j j i h j .
~12!
Inserting these in Eqs. ~11! and collecting terms to the same
order in j,h, linear equations for the Taylor coefficients
a i, j , b i, j are obtained. For the second-order coefficients the
solutions are
a 2,05s 1,2 ,
a 1,152s 2,1 ,
b 2,0523s 0,3 ,
a 0,253s 3,0 ,
b 1,1522s 1,2 ,
b 0,252s 2,1 .
x5 j 1s 1,2j 2 12s 2,1j h 13s 3,0h 2 1O@ u ~ j , h ! u 3 # ,
~13a!
y5 h 23s 0,3j 22s 1,2j h 2s 2,1h 1O@ u ~ j , h ! u # .
~13b!
2
3
Inserting this transformation in Eq. ~2!, we obtain
c 5a 0,2h 2 1a 3,0j 3 1 ~ a 2,126a 0,2s 0,3! j 2 h
1 ~ a 1,224a 0,2s 1,2! j h 2 1 ~ a 0,322a 0,2s 2,1! h 3
1O@ u ~ j , h ! u 4 # .
~14!
The s i, j are free for us to choose. First, we see that s 3,0 does
not appear in Eq. ~14! and we arbitrarily set it to zero. Second, with the choices
s 2,15
a 0,3
,
2a 0,2
s 1,25
a 1,2
,
4a 0,2
s 0,35
a 2,1
,
6a 0,2
~15!
we eliminate a number of terms and get
c 5a 0,2h 2 1a 3,0j 3 1O@ u ~ j , h ! u 4 # .
~16!
If a 3,0Þ0, the only remaining cubic term of c is nondegenerate, and, as we shall see below, the streamline pattern
can be determined. If a 3,050, the critical point is nonlinear
degenerate to order three, and further terms in the expansion
of c must be computed.
To continue and simplify the quartic terms of c, we use
a generating function of the form
S5y j 1
(
i1 j53
s i, j y i j j 1
(
i1 j54
s i, j y i j j ,
N
i53
~18!
where
ã 3,05a 3,0 ,
a 22,1
4a 0,2
1O~ a 3,0! ,
~19b!
a 1,2
a 2,1
2a 3,1
1O~ a 3,0! ,
a 0,2
2a 0,2
~19c!
etc. In ã k,0 , the term O(a 3,0) denotes expressions linear in
a 3,0 and also depending on a i, j with i1 j,k. Hence, ã k,0 is
a k,0 plus nonlinear combinations of a i, j , where i1 j,k. The
source of the complexity of the expressions in Eqs. ~19! is
apparent from the coordinate transformations. The terms in
the transformation to some order n do not only modify terms
of c to the same order, but to higher orders as well. The
exact expressions are readily computed, but they are long
and complicated and shed no further light on the problem, as
no physical interpretation presents itself.
The normal form of order N is obtained by dropping the
O term in Eq. ~18!. By the use of the normal form it is now
very easy to find the local flow picture of the degenerate
configurations. A degeneracy of order N occurs if ã 3,0
5¯5ã N21,050, but ã N,0Þ0. Then the normal form of c is
c 5a 0,2h 2 1ã N,0j N .
~20!
Possible separatrices ~dividing streamlines! of the critical point are given by c 50, that is
h 56
A
2
ã N,0 N
j .
a 0,2
~21!
If N is odd this describes two streamlines that meet at the
critical point in a cusp. If N is even, two subcases occur: If
ã N,0 /a 0,2.0 there are no separatrices and the critical point is
a degenerate center; if ã N,0 /a 0,2,0 we obtain four separatrices, and this case is denoted a topological saddle.23 See
Fig. 2.
~17!
where the s i, j of order three are given by ~15! and the s i, j of
order four now are free parameters. The Taylor series of the
transformation ~12! must be determined to order three in the
same way as above, and by insertion in c, the s i, j are to be
chosen to eliminate as many terms as possible.
In the general case, to any finite order N, one can show
that it is possible to choose the transformation, such that
c 5a 0,2h 2 1 ( ã i,0j i 1O@ u ~ j , h ! u N11 # ,
ã 4,05a 4,02
ã 5,05a 5,01a 4,0
Hence, the transformation ~12! becomes
2
FIG. 2. Degenerate critical points in the flow field. ~a! Cusp. ~b! Topological
saddle. ~c! Center.
~19a!
C. Unfoldings of degenerate critical points
The patterns found in the previous section only occur
when the parameters take certain degenerate values. A small
change from these combinations will result in qualitatively
different patterns. To study the bifurcations close to the
simple linear degeneracy, we introduce small parameters,
e 1 5a 1,0 ,
e 2 5a 0,1 ,
e 3 5a 2,0 ,
e 4 5a 1,1 ,
~22!
which all take the value zero when the origin is a critical
point with a simple linear degeneracy.
We proceed as before, but now include the small parameters in the transformations. We briefly sketch how to simplify quadratic terms. We choose a generating function,
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M. Bro” ns and J. N. Hartnack
Phys. Fluids, Vol. 11, No. 2, February 1999
S5y j 1
n
s i, j,k,l,m,n y i j j e k1 e l2 e m
(
3 e4 ,
i1 j1k1l1m1n53
~23!
and the new coordinates are found through the relations
x5
]S
,
]y
h5
]S
.
]j
317
~24!
Solving to find the transformation and inserting in the
streamfunction, we obtain
c 5 m 1 1 m 2 j 1 m 3 h 1 m 4 j 2 1 ~ 2s 2,1,0,0,0,024a 0,2s 0,2,1,0,0,0! e 1 j h 1 ~ 22s 1,2,0,0,0,024a 0,2s 0,2,0,1,0,0! e 2 j h
24a 0,2s 0,2,0,0,1,0e 3 j h 1 ~ 124a 0,2s 0,2,0,0,1 ! e 4 j h 1a 3,0j 3 2 ~ 4a 0,2s 1,2,0,0,0,02a 1,2! j h 2 1 ~ 26a 0,2s 0,3,0,0,0,01a 2,1! j 2 h
1 ~ 22a 0,2s 2,1,0,0,0,01a 0,3! h 3 1O~ 4 ! ,
~25!
where the m’s are new small parameters that are algebraic
combinations of the original e’s and O~4! denotes terms to
order four in both coordinates and parameters. By choosing
s 1,2,0,0,0,05
a 1,2
,
4a 0,2
s 0,3,0,0,0,05
s 0,2,0,0,0,15
1
,
4a 0,2
s 0,2,0,1,0,052
s 0,2,1,0,0,05
a 0,3
4a 20,2
a 2,1
,
6a 0,2
a 1,2
8a 20,2
s 2,1,0,0,0,05
a 0,3
,
2a 0,2
,
,
and setting the remaining coefficients in the generating function S equal to zero, the result is
S
c 5 e 12
S
D
a 1,2e 1 a 2,1e 2 2
1 e 31
2
j 1ã 0,2h 2 1a 3,0j 3 1O~ 4 ! ,
4a 0,2 2a 0,2
~26!
with
ã 0,25a 0,22
a 0,3e 2
.
2a 0,2
~27!
Note that all mixed terms in j and h have been removed.
If a 3,0Þ0, a further simplification is possible. The term of the
next-highest degree in a univariate polynomial can always be
removed by a suitable translation of the origin. This is
achieved by replacing j by j 1 j 0 and h by h 1 h 0 , where
j 05
2a 2,1e 2 2a 1,2e 1 24 e 3 a 0,2
,
12a 0,2a 3,0
h 05
e 2 a 0,2
.
a 0,3e 2 22a 20,2
~28!
A final simplification can be obtained by first dividing c
by 2ã 0,2—corresponding to scaling the time—and then scale
j by the substitution
S D
2ã 0,2
j→
3a 3,0
1/3
j,
~29!
N
f ~ x !5
( c ix i,
i51
where
s5
H
c N21 50,
~31!
21, for N even and a 0,2 /ã N,0,0,
11, for N even and a 0,2 /ã N,0.0 or N odd
and c i , i51,...,N22 are transformed small parameters.
The two subcases for N even stem from the scaling ~29!,
which then is
j→
U U
2ã 0,2
Nã N,0
1/N
j.
~32!
If a negative coefficient to the j N term occurs, c is replaced
by 2c and s 521 results.
Some general features of the normal form are evident.
First, c is an even function of y, and all resulting streamline
patterns will have reflectional symmetry in the x axis. The
differential equations for the streamlines are
ẋ5 s y,
ẏ52 f 8 ~ x ! .
~33!
All critical points lie on the x axis and are found as solutions
to f 8 (x)50. That is, there are, at most, N21 critical points.
At a critical point, the Jacobian is
S
0
s
2 f 9~ x !
0
D
,
~34!
and we immediately obtain that the critical point is a center if
s f 9 (x).0 and a saddle if s f 9 (x),0. The vorticity is
to obtain
c 5 12 h 2 1c 0 1c 1 j 1 13 j 3 1O~ 4 ! ,
s
c N5 y 21 f ~ x !,
2
1
c N5 ,
N
e 2 e 4 a 1,2e 22 a 0,3e 1 e 2
1
2
j 1 e 2h
2a 0,2 4a 20,2
2a 20,2
D
where c 0 , c 1 are new small parameters depending on
e 1 ,..., e 4 . Since the streamlines lie on isocurves of c we can
further omit the c 0 term.
If a 3,0 is also small, that is, we are close to a nonlinear
degeneracy of order four or more, terms of higher order must
be computed. In the general case one obtains, relabeling the
coordinates back to x, y, the following.
Theorem II.1: Let a 1,0 , a 0,1 , a 2,0 , a 1,1 , and
ã 3,0 ,¯ ,ã N21,0 be small parameters. Assuming the nondegeneracy conditions a 0,2Þ0,ã N,0Þ0 a normal form of order
N for the streamfunction (2) is
~30!
v 5¹ 2 c N 5 s @ s f 9 ~ x ! 11 # ,
~35!
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318
M. Bro” ns and J. N. Hartnack
Phys. Fluids, Vol. 11, No. 2, February 1999
FIG. 3. Bifurcation diagram for the normal form of order 3, Eq. ~38!.
which always has the sign of s at a center. Thus, the vortices
or eddying structures that occur all have the same sense of
rotation.
If f 9 (x)50, the critical point is degenerate and bifurcation occurs. The type of a degenerate point (x c ,0) can easily
be found by expanding the c in that point,
s
1
c 5 y 2 1 f ~ x c ! 1 ~ x2x c ! 3 f - ~ x c !
2
6
1
1¯1 ~ x2x c ! n f ~ n ! ~ x c ! .
n!
~36!
Separatrices are found as solutions to c 5 f (x c ) and it follows that the type of the critical point depends on the first
nonvanishing derivative of f (x). That is, if the order of the
first nonvanishing derivative is odd the degenerate point is a
cusp and if the order is even the critical point is either a
center ~s51! or a topological saddle ~s521!.
In addition to local bifurcations associated with degenerate critical points, there is also the possibility of global
bifurcations. These occur at parameter values where critical
points are connected by heteroclinic trajectories. This requires c to attain the same value at the critical points. The
algebraic condition for this is
f ~ x 1 ! 5 f ~ x 2 ! 5¯5 f ~ x p ! and
f 8 ~ x 1 ! 5 f 8 ~ x 2 ! 5¯5 f 8 ~ x p ! 50,
~37!
for all the p critical points (x i ,0) on the heteroclinic cycle.
c 3 5 y 1c 1 x1 x .
1 3
3
~38!
For c 1 50 the origin is a cusp. The critical points are found
from
f 8 ~ x ! 5c 1 1x 2 50,
~39!
which has solutions if c 1 ,0 given by x56 A2c 1 . Here we
have f 9 (x)52x, showing that the two critical points are a
saddle and a center. The unfolding is shown in Fig. 3. This is
the simplest possible way critical points can be created, and
this bifurcation occurs in a large variety of flow problems.
We show some examples of this in Sec. V.
32c 32 127c 21 50,
~42!
describing a bifurcation curve in the (c 1 ,c 2 ) parameter
plane. At the degenerate critical points, we have f - Þ0,
showing that they are cusps. Thus, as the bifurcation curve is
crossed, two critical points are created or destroyed, as
shown in Fig. 4. A simple computation shows that there is
one critical point when 32c 32 127c 21 .0, and three critical
points when 32c 32 127c 21 ,0. Their type depend on s.
For s 521, there can be two saddles, and a possibility
for heteroclinic bifurcation. A straightforward computation
shows that the conditions ~37! are fulfilled when c 1 50,
c 2 ,0. The process sketched in Fig. 4~a! can be viewed as an
eddy obtaining an inner substructure.
3. Normal form of order 5
~43!
For c 1 5c 2 5c 3 50 the origin is a cusp. Again, local bifurcations occur for
f 8 ~ x ! 5c 1 12c 2 x13c 3 x 2 1x 4 50,
f 9 ~ x ! 52c 2 16c 3 x14x 3 50.
~44!
Combining these, one obtains
x 4 1c 3 x 2 2 13 c 1 50.
~45!
It is convenient to use the discriminant D of this equation as
a parameter rather than c 1 ,
D5c 23 1 34 c 1 .
c 22 5 12 ~ 2c 3 6 AD !~ 2c 3 6 AD ! 2 ,
For N54, the streamfunction is
The condition for bifurcation is
By eliminating x56 A22c 2 /3 we find that bifurcation occurs for
~46!
When D>0 local bifurcation occurs for
2. Normal form of order 4
s
c 4 5 y 2 1c 1 x1c 2 x 2 1 14 x 4 .
2
f 9 ~ x ! 52c 2 13x 2 50. ~41!
c 5 5 12 y 2 1c 1 x1c 2 x 2 1c 3 x 3 1 15 x 5 .
For N53, the streamfunction is
2
f 8 ~ x ! 5c 1 12c 2 x1x 3 50,
For N55 the streamfunction becomes
1. Normal form of order 3
1
2
FIG. 4. Bifurcation diagram for the normal form of order 4, Eq. ~40!. ~a!
s51, ~b! s521.
~40!
~47!
for D,0 no local bifurcation occurs. The conditions for global bifurcation are
f ~ x1!5 f ~ x2!,
~48a!
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M. Bro” ns and J. N. Hartnack
Phys. Fluids, Vol. 11, No. 2, February 1999
319
The linearization of the streamline equations is now
SD S D S
0
2a 0,2
ẋ
a 0,1
5
1
0
ẏ
22a 2,0
0
DS D
x
,
y
~52!
so the origin is a critical point if a 0,150, which is a center if
u J u 54a 2,0a 0,2 is positive and a saddle if u J u is negative. A
degeneration occurs if u J u 50 and, as before, we consider
only the case when J is not the zero matrix.
We will again use normal form transformations to simplify the streamfunction at and close to the degeneracy, and
the algebra proceeds as in Sec. II, with the exception that we
now require that the transformations conserve the symmetry.
Denoting the new coordinates ~j,h!, this means that
j ~ 2x,y ! 52 j ~ x,y ! ,
FIG. 5. Bifurcation diagram for the normal form of order 5, Eq. ~43!. ~a!
D,0. ~b! D50. ~c! D.0.
f 8 ~ x 1 ! 5 f 8 ~ x 2 ! 50,
~48b!
f 9 ~ x 1 ! ,0∧ f 9 ~ x 2 ! ,0,
~48c!
for x 1 Þx 2 . By eliminating x 1 and x 2 from Eqs. ~48a!–~48b!
by the use of resultants,29 one finds that global bifurcations
occur for
3D ~ 9D
2
266Dc 23 1161c 43 118c 3 c 22 !
5392c 63 1142c 33 c 22 29c 42 .
~49!
Notice that the constraint ~48c! is not included here and only
a part of the curve ~49! describe global bifurcation points.
The complete bifurcation diagram is shown in Fig. 5.
III. PATTERNS IN SYMMETRIC FLOWS
In this section we will investigate streamline topologies
under the assumption of reflectional symmetry. We take the
axis of symmetry to be the y axis, and the streamfunction is
then an even function of x,
c ~ 2x,y ! 5 c ~ x,y ! ,
~50!
and the expansion ~2! becomes
`
c5
(
i1 j51
a 2i, j x 2i y j .
~51!
h ~ 2x,y ! 5 h ~ x,y ! ,
~53!
and it follows that the generating function ~23! must be an
odd function of j.
Two subcases must be distinguished. If a 0,250 and
a 2,0Þ0 (“u50, “uÞ0) the normal form computations lead
to exactly the same result as in Theorem II.1, but with x and
y interchanged. Thus, the symmetry that always occurs in the
general normal form covers this case as well, and the assumption of symmetry give rise to no further limitations in
the possible streamline patterns and their bifurcations.
In the other subcase a 2,050 and a 0,2Þ0 (“ v 50, “u
Þ0), the normal form procedure gives rise to the following.
Theorem III.1: Let a 0,1 , a 2,0 and ã 4,0 , ã 6,0 ,...,
ã 2 @ N/2# 22,0 be small parameters. Assuming the nondegeneracy conditions a 0,2Þ0, ã 2 @ N/2# ,0Þ0 a normal form of order
N for the streamfunction (51) is
s
c N5 y 21 f ~ x !,
2
c 2 @ N/2# 5
@ N/2#
f ~ x !5
(
i51
c 2i x 2i ,
1
,
2 @ N/2#
~54!
where
s5
H
21,
11,
for a 0,2 /ã 2 @ N/2# ,0,0,
for a 0,2 /ã 2 @ N/2# ,0.0,
~55!
and c 2i , i51,...,@ N/2# are transformed small parameters.
In the theorem @•# denotes the integer part. The structure
of the normal form is exactly the same as for the general case
in Sec. II, only with no terms of odd order in x. Thus, we
reach the same general conclusions here but notice that the
origin is always a critical point and the total number of critical points is odd. In addition to the local bifurcation occurring when f 9 (x)50 and f 8 (x)50, global bifurcation takes
place when the streamfunction attains the same value at two
saddle points x 1 , x 2 that are not a symmetric pair, that is,
when u x 1 u Þ u x 2 u .
The algebraic manipulations involved in obtaining the
bifurcation diagrams in the symmetric case follows closely
the procedure in Sec. II. We omit the details, and only show
the resulting diagrams for codimension up to three in Figs.
6–9.
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320
M. Bro” ns and J. N. Hartnack
Phys. Fluids, Vol. 11, No. 2, February 1999
In general, the equation associated with the term x i y j is
~ j14 ! !
~ i12 ! ! ~ j12 ! !
a i, j14 12
a i12,j12
j!
i!
j!
1
FIG. 6. Bifurcation diagram for the symmetric normal form of order 4. ~a!
s 521. ~b! s 51.
IV. THE ROLE OF THE VORTICITY TRANSPORT
EQUATION IN STEADY FLOWS
The patterns we have discussed until now are based only
on the existence of a streamfunction c for which only incompressibility is assumed. Thus, all patterns can be realized,
since they can be used as initial conditions for a temporal
development. Here we address the question as to whether the
patterns can appear in steady flows.
For a 2-D viscous flow, the Navier–Stokes equations can
be combined into the vorticity transport equation by eliminating the pressure. In a streamfunction formulation for
steady flow, the equation is
n ¹ 4c 5
]c ]
]c ]
~ ¹ 2c !2
~ ¹ 2c !,
]y ]x
]x ]y
~56!
where n is the kinematic viscosity. Inserting the expansion
~2! and collecting terms of the same order in x, y yields a
series of algebraic equations for the a i, j . The equations to
order zero and one are
x 0 y 0 : 8 n ~ a 2,213a 4,013a 0,4!
52a 0,1~ 3a 3,01a 1,2! 22a 1,0~ 3a 0,31a 2,1! ,
~57a!
x 1 y 0 : 24n ~ 5a 5,01a 3,21a 1,4!
526a 1,0~ a 1,31a 3,1! 14a 0,1~ a 2,216a 4,0!
12a 1,1~ a 1,213a 3,0! 24a 2,0~ a 2,113a 0,3! , ~57b!
x 0 y 1 : 24n ~ 5a 0,51a 2,31a 4,1!
56a 0,1~ a 1,31a 3,1! 24a 0,1~ a 2,216a 0,4!
22a 1,1~ a 2,113a 0,3! 14a 0,2~ a 1,213a 3,0! . ~57c!
~ i14 ! !
a i14,j 5 f i, j ,
i!
~58!
where f i, j are nonlinear combinations of coefficients a l 8 ,m 8 ,
with l 8 1m 8 ,i1 j14. The structure of these equations is
clear: For a given order k the set of equations for x i y j with
i1 j5k consists of a linear combination of a l,m , with l
1m5k14 on the left hand side and a nonlinear combination of a l 8 ,m 8 with l 8 1m 8 ,k14 on the right-hand side.
Hence, these equations give linear constraints on the coefficients of a certain order expressed in terms of lower-order
coefficients.
There are k11 equations in the set to order k but k15
unknown coefficients of order k14. Thus, the system is not
fully determined. However, it is easy to see that the system
has rank k11 and that a k,0 can be chosen as one of the four
free parameters. Hence the equations give no information
about a k,0 which enters linearly in the decisive parameter ã k,0
defined in Eqs. ~19!, which consequently is left undetermined.
Furthermore, there is no information about coefficients
of order less than four, so we conclude that the vorticity
transport equation does not give any constraints on steady
streamline patterns and their bifurcations.
In the limit n →` we have Stokes flow, and Eq. ~56! is
replaced by
¹ 4 c 50,
~59!
and the right hand side of Eq. ~58! becomes zero. Since the
arguments above were based entirely on the linear part of the
Navier–Stokes equations we reach the same conclusion for
Stokes flow: All topologies and bifurcations described in
Sec. II can occur in steady Stokes flow.
Steady flow was also considered by Bakker23 in his
analysis of flow close to a wall. However, since the constraints from the Navier–Stokes equations were inserted before the topological analysis, their role was not clear. In an
analysis similar to the present one, Hartnack25 found that
FIG. 7. Bifurcation diagram for the symmetric normal form of order 6. ~a! s 521. ~b! s 51.
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M. Bro” ns and J. N. Hartnack
Phys. Fluids, Vol. 11, No. 2, February 1999
FIG. 8. Bifurcation diagram for the symmetric normal form of order 8 with
16
s 51. ~a! D,0. ~b! D50. ~c! D.0. Here, D516c 26 2 3 c 4 .
steadiness poses no topological constraints for the unfolding
of a simple degeneracy at a wall. However, in a low-order
case with a nonsimple degeneracy some patterns were ruled
out for steady flow.
For flows away from walls, a systematic study of the
role of steadiness for nonsimple degeneracies has not been
performed. However, from the vorticity transport equation
~56! it follows that only coefficients to terms of fourth order
or higher in the streamfunction are constrained. Further nonsimple degeneracies with topologies determined by fourthorder terms or higher in the streamfunction are of codimension 4 or more. This means that such degeneracies are not
very common in flows, and the question of whether steadiness imposes a restriction on the topology of these degeneracies or not becomes rather academic.
Let us now consider the more subtle case of flows with
reflectional symmetry. Without loss of generality the symmetry line will be taken as x50, and the streamfunction now
has the expansion ~51!, all terms being even in x. Equations
~57! now become
x 0 y 0 : a 2,213a 4,013a 0,450,
~60a!
x 1 y 0 : 052a 0,1~ a 2,216a 4,0! 22a 2,0~ a 2,113a 0,3! , ~60b!
x 0 y 1 : 5a 0,51a 2,31a 4,150.
~60c!
321
FIG. 9. Bifurcation diagram for the symmetric normal form of order 8 with
s521. ~a! D,0. ~b! D50. ~c! D.0.
A problem for the two equations of order k51 for which the
unknowns are the coefficients of order five is evident. Solutions exist only if the nonlinear constraint, Eq. ~60b!, is fulfilled. Furthermore, if we continue to the equations for higher
orders k we encounter a similar problem for each equation
with an odd power of x. This gives rise to infinitely many
nonlinear constraints that involve low-order terms in a very
complicated pattern.
We leave the question on the solution of these constraints unresolved here, and turn to the case of Stokes flow
with symmetry. Again we get some equations with zero lefthand sides, but now the right-hand sides are all zero, and we
are back to a structure similar to the unsymmetric case. The
nonzero equations of order k constitute an underdetermined
homogeneous linear system, and only the coefficients of
even order have topological interest according to Theorem
III.1. It is not difficult to see that the a 2n,0 can be chosen as
free parameters, and again the ã 2n,0 are undetermined.
Thus, for symmetric Stokes flow steadiness poses no topological constraint. Whether it distinguishes Stokes flow
from Navier–Stokes flow is an open problem.
V. APPLICATIONS
A. Stokes flow in a rectangular cavity
As an application of the flow patterns found, we investigate Stokes flow in a rectangular cavity with length a and
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322
M. Bro” ns and J. N. Hartnack
Phys. Fluids, Vol. 11, No. 2, February 1999
FIG. 10. The geometry of Stokes flow in a rectangular cavity.
height b. The flow is driven by sliding the top and the bottom
of the box. These two boundaries are allowed to move independently of each other at constant speed u t and u b , respectively. The two sidewalls are fixed. The flow is governed by
the Stokes equation,
¹ 4 c 50.
~61!
The axes of the system and the streamfunction are nondimensionalized through
x5aX,
y5bY ,
C5
c
,
bu b
~62!
which applied to Eq. ~61! gives the nondimensional equation
H4
] 4C
] 4C
] 4C
2
12H
1
50,
]X4
] X 2] Y 2 ] Y 4
b
with H5 , ~63!
a
which has to be satisfied in a unit square. The situation is
sketched in Fig. 10.
With the given geometry there are two parameters which
can be varied independently. The height to length ratio,
b
H5 ,
a
~64!
and the ratio of the top and bottom wall velocities,
S5
ut
.
ub
~65!
By varying these two one obtains different flow topologies.
The sketched cavity flow has been studied recently numerically by Kelmanson and Lonsdale.26 They solved the
problem using a numerical integral equation method.
Shankar30 also studied the cavity flow, though he considered
only the motion of one endwall and used a numerical method
using a truncated eigenfunction expansion. Here we show
solutions obtained by a finite difference scheme developed
by Altas et al.31 This scheme uses not only the streamfunction as unknown but also the first derivatives. This expands
the number of unknowns to three. This increase in unknowns
is compensated by the exact implementation of the boundary
conditions instead of introducing approximations at boundaries. Also, this scheme gives direct access to the critical
points since these can be found as intersections of the isoclines c X 50 and c Y 50.
Since the system has two independent parameters, one
should not expect to observe degenerate streamline configu-
FIG. 11. Streamline patterns for Stokes flow in double lid driven cavity, for
different values of the velocity ratio S: ~a! S520.70; ~b! S520.90; ~c!
S521.00, ~d! S521.10; and ~e! S521.30.
rations of codimension more than 2. Using a coordinate system fixed at the center of the box ~as shown in Fig. 10! the
system has reflectional symmetry in the Y axis and the domain of the problem is bounded giving closed streamlines.
The bifurcation diagrams that are appropriate for this system
are consequently Fig. 4~a! ~with x and y interchanged!, Fig.
6, and Fig. 7~b!.
In Fig. 11 a series of streamline patterns is displayed for
different values of the velocity ratio S and a fixed height to
length ratio of unity. By variation of the velocity ratio, one
encounters exactly the bifurcations of Fig. 4~a!. In Fig. 11~a!
only one center point is present, by decreasing S to around
20.90 a cusp critical point appears above the center point.
By decreasing S below 20.90 this cusp critical point evolves
into a saddle center configuration in ~c!, giving one large
vortex with an inner ‘‘figure-eight’’-shaped structure.
Through a further decrease in S the top vortex grows in
strength, annihilating the bottom vortex through a second
cusp bifurcation at S521.10. The process thus observed
corresponds to varying c 1 across the cusp-shaped bifurcation
curve along a curve with c 2 ,0 in Fig. 4~a!.
If this experiment was repeated at a slightly different
value of H, the same behavior is expected, but with the specific values of S, where the bifurcations occur also slightly
altered, corresponding to moving along a slightly different
curve in the (c 1 ,c 2 ) parameter space. Hence, by varying the
two physical parameters H, S, a bifurcation diagram corresponding to Fig. 4 will result. It may be deformed but the
cusp shape will persist.
Other bifurcations are present in the system. Again, for
H51 Kelmanson and Lonsdale26 observe in the range
0.05,S,0.25 the metamorphosis of a ‘‘double eddy’’ ~that
is, a ‘‘figure-eight’’! into a ‘‘treble eddy’’ and further into a
single eddy. The patterns are shown in their Figs. 5, 6. This
sequence corresponds exactly to the bifurcations in Fig. 7~b!
when c 2 is changed from negative to positive along a curve
with c 4 ,0, crossing both bifurcation curves below the c 2
axis. We conclude that the complete bifurcation diagram of
Fig. 7~b! is expected to be present in a H, S bifurcation
diagram.
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M. Bro” ns and J. N. Hartnack
Phys. Fluids, Vol. 11, No. 2, February 1999
323
FIG. 13. Flow patterns that do not appear in the unfoldings of a simple
linear degenerate critical point: Critical points in a triangle and saddles with
a single heteroclinic connection.
FIG. 12. The flow structures listed by Gaskell et al. ~Ref. 27! as being
possible in a Stokes flow in a cavity.
Hence, the present theory can guide the construction of
pattern bifurcation diagrams. This approach has been used by
applying the corresponding bifurcation diagrams for axisymmetric flow18 to map the topologies in the steady vortex
breakdown.19,32
tern ~f! can bifurcate from pattern ~b! through the creation of
off-axis cusp points.
An explanation for the missing bifurcation of critical
points off the axis may be found in the shape of the flow
domain. The existence of the series solution hinges on the
free surfaces being radial lines. Special topological properties may be associated with these solutions. These can be
destroyed while keeping the reflectional symmetry, for instance, by changing the amount of fluid such that the annulus
is not exactly half-filled. Hence, if the height of the free
surface is varied as a third parameter, it may well be possible
to encounter a more complex set of flow topologies and bifurcations, and the subset found in Ref. 27 may turn out to be
quite special.
B. Stokes flow between concentric cylinders
Gaskell et al.27 have considered Stokes flow in a halffilled annular region between rotating concentric cylinders.
The problem is conveniently described in polar coordinates,
where the lines with u 56 p /2 are the free surfaces. The
resulting boundary value problem is characterized by two
dimensionless parameters: the ratio R̄ of the radii of the cylinders and the ratio S of the peripheral speeds of the cylinders. A series solution is obtained assuming reflectional symmetry across the centerline u 50, and a rich set of eddying
motions is found by evaluating the solution numerically.
A basic flow in the system consists of a vortex with its
center on the centerline. As the parameters are varied (R̄
being the most important!, this can bifurcate to a saddle and
two off-axis centers, like in Fig. 6~b! for decreasing values of
c 2 . A further variation of parameters can change the saddle
back to a center and at the same time create two off-axis
saddles, as in Fig. 6~a! for decreasing values of c 2 . These
two bifurcations occur repeatedly, resulting in more and
more complex patterns as the number of critical points rise.
Gaskell et al. list a number of flow structures ~reproduced in Fig. 12! that they consider to be possible for the
flow, though only the patterns ~a!, ~b!, ~c!, and ~h! were
realized.
From the present analysis it is clear that the patterns ~d!
and ~g! are not to be expected numerically. They are structurally unstable due to the multiple saddle connections and a
slight change of parameters can break some of the connections. For example, pattern ~d! occurs in Fig. 9~c!, but along
a bifurcation surface only.
The lack of patterns ~f!, ~i! is more surprising. Since
there are two control parameters in the system, one would
expect bifurcations with codimension up to two to occur generically. Such bifurcations are exactly those described in
Fig. 7, and if the ‘‘unfinished’’ separatrices in Fig. 7~a! are
joined around far off-axis centers, one finds indeed that pat-
VI. CONCLUSIONS
The use of normal forms in the analysis of flow patterns
gives a significant simplification and allows an easy determination of the possible degenerate patterns and their bifurcations, even to quite high codimension. The structure of the
normal form depends only on the Jacobian matrix of the
velocity field at a degenerate critical point. In the case considered here, with a simple linear degeneracy, all critical
points lie on a line in the normal form coordinates, and configurations with critical points in a triangle cannot appear.
Also, a single heteroclinic connection between saddles violates the inherent symmetry of the normal forms. See Fig. 13.
All vortices also have the same sign and an interaction between opposite vortices as shown in Fig. 14 is not captured.
Hence, an understanding of a number of interesting and
important patterns require an analysis of degeneracies with a
zero Jacobian matrix. Unfortunately, a direct use of the normal form approach in this case gives no simplifications at all.
A systematic elimination of unimportant terms is difficult,
reflecting the much richer set of structures that is possible
here. In a similar situation for the flow close to a wall,
Hartnack25 have obtained simplification of low-order terms.
The same approach may be used for the present case.
The topological classifications obtained in this paper
hinge on the coefficients of the normal forms. In principle,
these can be expressed in terms of the Taylor expansion coefficients a i, j of the streamfunction, which, in turn, can be
interpreted physically in terms of the stress tensor, as indi-
FIG. 14. The interaction of two vortices with opposite rotation.
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324
Phys. Fluids, Vol. 11, No. 2, February 1999
cated in Sec. II A. However, apart from the coefficients of
the very lowest orders, this algebraic connection is a very
complicated one that we have not even attempted to write
out explicitly. This is not a flaw of the approach used here
but a basic property of fluid patterns. In consequence, there
are, in general, no simple physical quantities that can be used
as indicators for pattern bifurcations. Thus, it is inherently
difficult to single out the physical mechanisms that are responsible for pattern changes, except in the simplest cases.
While a topological classification in itself cannot solve a
problem in fluid mechanics, it provides guidelines for identification of structures in any flow. We have indicated how
this could be done for two cases of Stokes flow. A more
detailed study of the structures in the Stokes flow in a cavity
is in preparation.
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